University of Utah, Department of Mathematics May 2016, Algebra Qualifying Exam

Let \(G\) be a group (not necessarily finite), and suppose that \(H\) is a subgroup of index \(n\). Show that there is a normal subgroup \(N\) of \(G\) with \(n!\ge [G:N]\ge n\).

Determine, up to isomorphism, the number of groups of order 70.

Let \(p\) be a prime integer, and \(G\) a group in which \(g^p\) is the identity for each \(g\) in \(G\). Show that \(G\) must be abelian if \(p=2\). Give an example where \(G\) is not abelian.

Let \(R=\mathbb {Q}[x]\), and let \(M\) be the cokernel of the map \(R^2\longrightarrow R^3\) given by the matrix \[ \begin {pmatrix} x & 0\\ x & x^2\\ 1 & 1 \end {pmatrix}. \] Write \(M\) as a direct sum of cyclic \(R\)-modules.

Compute the characteristic polynomial, the minimal polynomial, and the Jordan form of the matrix \[ \begin {pmatrix} 3 & 1 & -1\\ 2 & 2 & -1\\ 2 & 2 & 0 \end {pmatrix}. \]

Let \(K[x]\) be a polynomial ring over a field \(K\), and let \(n\) be a positive integer. Classify, up to isomorphism, all finitely generated modules over the ring \(K[x]/(x^n)\).

Factor \(11x^5-11x^4+14x^2-21x+7\) into irreducible polynomials in \(\mathbb {Q}[x]\).

Prove that the polynomial \(x^5-x-1\) has no root in \(\mathbb {F}_9\), and that it is irreducible over \(\mathbb {F}_3\). Determine the integers \(n\) for which \(x^5-x-1\) is irreducible over \(\mathbb {F}_{3^n}\).

Show that \(K=\mathbb {Q}(\sqrt {1+\sqrt {3}})\) is not Galois over \(\mathbb {Q}\), and compute \([K:\mathbb {Q}]\).

Let \(p\) be a prime integer, and set \(f(x)=x^{p-1}+x^{p-2}+\cdots +x+1\). Suppose a prime integer \(q\) divides \(f(a)\) for some integer \(a\), prove that either \(q=p\) or \(q\equiv 1\bmod p\).

Use this to prove that the arithmetic sequence \(1,1+p,1+2p,\ldots \) contains infinitely many prime integers.